Systems and methods for locating and imaging proppant in an induced fracture

ABSTRACT

Born Scattering Inversion (BSI) systems and methods are disclosed. A BSI system may be incorporated in a well system for accessing natural gas, oil and geothermal reserves in a geologic formation beneath the surface of the Earth. The BSI system may be used to generate a three-dimensional image of a proppant-filled hydraulically-induced fracture in the geologic formation. The BSI system may include computing equipment and sensors for measuring electromagnetic fields in the vicinity of the fracture before and after the fracture is generated, adjusting the parameters of a first Born approximation model of a scattered component of the surface electromagnetic fields using the measured electromagnetic fields, and generating the image of the proppant-filled fracture using the adjusted parameters.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of and priority to U.S. ProvisionalPatent Application No. 61/948,169 filed Mar. 5, 2014, which is herebyincorporated by reference in its entirety.

STATEMENT OF GOVERNMENTAL INTEREST

The United States Government has rights in this invention pursuant toContract No. DE-AC04-94AL85000 between the United States Department ofEnergy and Sandia Corporation, for the operation of the Sandia NationalLaboratories.

BACKGROUND

As the population of the world increases, efficient mechanisms forobtaining sources of energy, including natural gas, oil and geothermalreserves are continuously being investigated. One exemplary techniquefor obtaining access to natural gas, oil and geothermal reserves isknown as hydraulic fracturing. Hydraulic fracturing is the process ofinitiating and subsequently propagating a fracture in a geologicformation through utilization of fracturing fluid. To create thefracture in the geologic formation, a drill is employed to create a wellbore that reaches depths of several thousand feet (until a desiredgeologic formation is reached). A well casing is placed in the wellbore. The well casing is typically composed of steel. The well casing iscemented in place to stabilize the well casing with respect to theEarth.

Hydraulic fracturing is commonly employed to enhance the fluid flowpermeability of shale geologic formations for petroleum (oil and/ornatural gas) and geothermal energy production. Subsequent to the wellcasing being cemented in place, a fracturing fluid pumped down the wellbore and through perforations in the well casing at a pressure that isin excess of the fracture gradient of the geologic formation. Suchpressure causes the geologic formation to fracture. Pumping of thefracturing fluid down the well bore is continued to extend the fracturefurther into the formation. As the fracture extends, a proppant is addedto the fracture fluid and pumped down the well bore and into thefracture, thereby propping the fracture open when pumping of thefracture fluid ceases. This causes the geologic formation to becomepermeable via the fracture, thereby allowing natural gas or oil to beextracted from the geologic formation. Hydraulic fractures can beinduced using vertical, horizontal and/or slanted wells. This process iscommonly referred to as hydraulic fracturing.

Because a typical fracture occurs thousands of feet beneath the surfaceof the Earth and because a fracture can extend from the well bore in avariety of directions and orientations, it is difficult to determine thelocation of a fracture within the geologic formation. Modelingtechniques have been developed in which, prior to a hydraulic fracturingoperation, electromagnetic fields at the surface of the Earth resultingfrom an application of an electric current to various hypotheticalfractures through the well bore are calculated. Following the hydraulicfracturing operation, an electromagnetic field measured at the surfaceof the Earth is used to select from the various hypothetical fractures.Although these modeling techniques have been successful in helping tolocate induced fractures, they are limited by the number and accuracy ofthe hypothetical fractures used to compute the predicted fields.

In contrast to these forward modeling approaches, an inverse modelingsolution in which measured fields are used to infer the location andorientation of the fracture, rather than simply selecting from a groupof hypothetical fractures, has long been desired. However, in order toinfer fracture location, orientation, geometry, etc., from measured EMfield data using conventional techniques, EM field data must be computedand compared with the measured field data many times. Because thecomputation time for computing a model can be long, it has beeneconomically and practically infeasible to wait for this type of inversemodel computation after a hydraulic fracturing operation and before theextraction of the natural gas, oil or geothermal resources. Noeconomically and computationally feasible inverse modeling solution hastherefore been forthcoming.

It would therefore be desirable to provide improved systems and methodsfor evaluating well hydraulic fracturing and completion techniquesuseful in extracting natural gas, oil and geothermal reserves from ageologic formation.

SUMMARY

The following is a brief summary of subject matter that is described ingreater detail herein. This summary is not intended to be limiting as tothe scope of the claims.

Described herein are various technologies pertaining to modeling theproperties of a fracture in a geologic formation. Properties includefracture size and geometry. These properties are determined or inferredfrom determined proppant and/or fracture properties. The fracture may beused to extract natural gas, oil and geothermal reserves from thegeologic formation

The disclosed methodology and associated modeling utilizeselectromagnetic (EM) energy scattered from a subsurface geological,geophysical or artificial feature of interest to produce ahighly-resolved three-dimensional (3D) image of the feature. The methodincludes producing an image or representation of a hydraulic fractureinduced in a subsurface geologic formation after the fracture has beeninjected with propping material (“proppant”) with favorable EMcharacteristics.

In various embodiments, properties such as the size, shape, location,orientation and/or extent of the proppant pack and fracture may bedetermined in an economically and computationally feasible inversemodeling operation. The modeling operation may be used to generate athree-dimensional image of proppant material within a fracture. As usedherein, the terms “proppant material” and “proppant” refer to materialthat includes many (e.g., thousands, millions or more) of individualproppant particles or elements.

According to the present invention, a model is disclosed that estimatesor calculates electromagnetic (EM) field values at the locations of oneor more sensors at or near the surface of the Earth and then adjustscalculated EM field values in the model's Earth volume based on measuredelectromagnetic field data gathered by the sensors before and after ahydraulic fracking operation. The model includes a geophysical model,which may be referred to as an “earth model,” of a volume of the Earththat includes at least a portion of a geologic formation and a wellbore. The geophysical model is a three dimensional representation of theEM medium properties of the volume of interest, including but notlimited to the geologic formation, well bore and casing, overburden, andthe surface of the earth over the volume.

The model includes a First Born Approximation (FBA) model component,which may be referred to herein as a First Born Approximation modelingprocedure, calculation or operation. The FBA model includes acalculation using measured EM fields taken before and after a fracturingoperation to adjust parameters in the FBA model. This method may bereferred to as a Born Scattering Inversion (BSI) operation.

The FBA is a mathematical approximation to Maxwell's equations (thegoverning equations of electromagnetism) that posits that the strengthof an electromagnetic wavefield scattered by a localized contrast inmaterial properties (e.g., a proppant-filled fracture in a geologicformation) is linearly related to the strength of the incidentelectromagnetic wavefield, and the magnitude of the contrast. Thiscontrast can be described at various locations using a set of adjustableparameters in an FBA model.

The FBA modeling operation includes two successive executions of anelectromagnetic modeling algorithm. In the first execution, incidentelectromagnetic fields, which may be referred to a primary EM fields, atone or more predetermined locations in the volume of the Earth arecalculated. In the second execution, each of the predetermined locationsare treated as an electromagnetic wavefield source (sometimes referredto herein as a Born scatterer or a Born scattering source) that isactivated by the previously computed incident electromagnetic fieldwaveforms. In the second execution, scattered EM fields, which may bereferred to as secondary electromagnetic fields, from the Bornscatterers at the locations of the sensors at the surface of the Earthare computed.

The adjustable parameters of the FBA model are scaling values thatdetermine a relation between the strength of the incidentelectromagnetic wavefield and the scattered electromagnetic field. Thescaling values correspond to scattering amplitudes of a plurality ofBorn scatterers. The scattered electromagnetic fields are calculatedusing initial values such as unit values for the adjustable parametersor an estimated value of an EM parameter.

EM field data are measured or obtained before and after fracturing andproppant placement. The difference between the field data before, duringand/or after the hydraulic fracturing operation is equivalent to ascattered EM field that is generated primarily by a change in theelectromagnetic properties of some locations within a volume of Earth.The change in the EM properties at some of these locations can be theresult of the presence of the proppant in the fracture. The differencebetween the measured field data before and after the hydraulicfracturing operation may therefore be compared to the calculatedscattered EM fields from the Born scatterers to determine proppant orproppant pack location.

The field data are then used to adjust parameters of the FBA model sothat the calculated scattered EM fields match the difference in themeasured EM fields to within a predetermined or actively determinedrange. In some embodiments, the adjusted values of the adjustableparameters are determined by the measured field data. In someembodiments, adjusting the parameters may include a linear inverseoperation that directly solves for the adjusted parameters. In anotherembodiment, adjusting the parameters may be by another technique oroperation that adjusts the strength of the parameters by, such as, butnot limited to EM migration, full waveform inversion, and Monte Carlotechniques. The adjusted parameters for each location may indicatewhether proppant is present at that location, because the change in theEM properties at the locations of the Born scatterers is due to thepresence (or lack) of proppant material at those locations.

In one illustrative example, an adjusted parameter that is equal to zerofor a Born scatterer may indicate that that Born scatterer is notlocated within a proppant-filled fracture or that a fracture is notpresent. An adjusted parameter that is different than zero for anotherBorn scatterer may indicate that that Born scatterer is located within aproppant-filled fracture. The values of the adjusted parameters maytherefore be used to determine a location, shape, size, extent, and/ororientation of proppant in the fracture. In a modeling operation inwhich a three-dimensional distribution of Born scatterers is used, thevalues of the adjusted parameters may be used to form athree-dimensional image of proppant within a fracture or elsewhere in awell system.

According to an embodiment, a system is provided that includes adatabase and a processor that calculates EM field values, receivesmeasured EM field data, and adjusts, based on the measured EM fielddata, the calculated EM field values. The database and processor mayinclude one or more databases and/or processors.

According to an embodiment, the database stores a geophysical model of avolume of Earth including a geologic formation and a well bore, a set ofBorn scatterer locations within the volume, an EM model for simulatingor calculating EM data, and a set of sensor locations. The processorcalculates the EM field values at the set of sensor locations using theEM model and the geophysical model, and receives measured EM field datagathered at the set of sensor locations to adjust EM parameters. Theelectromagnetic model includes a first Born approximation modelcalculating a magnitudes a plurality of Born scatterers at a set ofpredetermined locations within the volume.

According to another embodiment, the processor calculates the EM fieldvalues at a set of sensor locations by computing primary electric fieldvalues at the set of predetermined Born Scatterer locations within thevolume and computing secondary electric field values at the set ofsensor locations using the primary electric field values at the set ofBorn scatterer locations within the volume.

According to an embodiment, a method is provided that includesdetermining a plurality of scattered electromagnetic field values usinga model having adjustable parameters, performing a hydraulic fracturingoperation to create a fracture in a geologic formation, providing anelectromagnetically suitable proppant into the fracture, gathering,prior to the hydraulic fracturing operation, a plurality of measuredelectromagnetic field values at a first plurality of sensor locations,gathering, with the electromagnetically suitable proppant in thefracture, an additional plurality of measured electromagnetic fieldvalues at a second plurality of sensor locations, determining adifference between the plurality of measured electromagnetic fieldvalues and the additional plurality of measured electromagnetic fieldvalues, and modifying at least some of the adjustable parameters basedon a comparison of the difference between the plurality of scatteredelectromagnetic field values taken before and after fracturing. Themodel includes a first Born approximation model portion. According to anembodiment, the first plurality of sensor locations is the same as thesecond plurality of sensor locations. According to another embodiment,the first plurality of sensor locations is different than the secondplurality of sensor locations.

According to an embodiment, a system is provided that includes aconductive well casing in a well bore that extends from the surface ofthe Earth to a geologic formation within the Earth, an electricitysource conductively coupled to the conductive well casing, a proppant ina fracture in the geologic formation and conductively coupled to thewell casing, a plurality of sensors at a corresponding plurality ofsensor locations, wherein the sensors are configured to gatherelectromagnetic field data generated when a current is applied to theconductive well casing using the electricity source, and computingequipment that includes memory that stores a plurality ofelectromagnetic field values at the plurality of sensor locations basedon a first Born approximation model that includes a plurality ofadjustable scaling factors, and a processor configured to receive theelectromagnetic field data from the plurality of sensors, adjust theadjustable scaling factors based on the electromagnetic field data, andgenerate an image of the proppant using the adjusted adjustable scalingfactors.

According to an embodiment of the invention, a system and method forimaging a fracture is disclosed that includes field acquisition of EMdata, insertion of favorable proppant material within the fracture,computer processing and modeling of the recorded data, and ultimatelyconstructing a 3D image. Two sets of electromagnetic field data areacquired, before and after a hydraulic fracturing operation. Recorded EMdata consist of time series of components of the electric vectore(x_(r),t), the magnetic vector h(x_(r),t), or both, observed at a setof receiver locations x_(r).

According to an embodiment of the invention, the field data acquisitionmethod includes that the EM energy sources and sensors in both recordingexperiments occupy the same or similar positions. The other recordingconditions (e.g., EM energy source magnitudes, waveforms, andorientations; EM receiver sensitivities and orientations, recordingsystem filters and sampling characteristics, etc.) are maintainedidentical or as closely as possible between the two data collections.Extraneous sources of EM energy should be minimized and ambient noiselevels associated with the repeated experiments should be nearly thesame. The intention is that the difference in recorded EM data from thetwo experiments is attributable solely to the presence of theproppant-filled fracture. The method is not limited to any particularfield data acquisition configuration. Rather, EM energy sources andreceivers may be deployed on the earth's surface or within subsurfaceboreholes (vertical, horizontal, dipping). Line arrays, areal arrays, oreven volumetric arrays (as with multiple boreholes) of EM sources andreceivers may be utilized. However, proximity of sources and receiversto the target of interest will enhance recorded EM signal levels andprovide a better-constrained inversion result.

According to another embodiment of the invention, the EM energy sourcesand/or sensors used before and after the fracturing and proppantplacement are not located in the same positions, but the data isinterpolated to determine the difference in EM signals at the samegeophysical locations.

After the fracture is induced, it is injected with proppant materialpossessing EM characteristics that differ from the surrounding geologicformation. The salient EM properties are current conductivity (σ)electric permittivity (ϵ), and magnetic permeability (μ). The proppantmay be enhanced in any one, or any combination (including all threeconcurrently), of these properties with respect to the surroundingmedium. The stronger the EM material property contrast between theproppant and the geologic formation, the stronger will be the scatteredelectromagnetic wavefield recorded in the post-fracturing dataacquisition experiment.

The basic data utilized for imaging of a proppant-filled fractureconsists of the difference in EM data recorded by pre-fracture andpost-fracture field experiments. This difference data is directlymodeled (or simulated) by a numerical algorithm based on the First BornApproximation (FBA) applied to electromagnetic wavefieldpropagation/diffusion. The FBA posits that the strength of an EMwavefield scattered by a localized contrast in material properties (likea proppant-filled fracture) is linearly related to the strength of theincident EM wavefield, and the magnitude of the parameter contrast. FBAmethodology involves two successive executions of an EM modelingalgorithm. In the first run, the EM wavefield incident onto a 3D targetis recorded by a finite set of multi-component EM receivers distributedover the volume occupied by the target. For the case of a hydraulicfracture initiated at a known point in a borehole, a reasonable estimateof the location and extent of this volume is readily made. In the secondFBA modeling run, the receivers are considered EM wavefield sourcesactivated by the incident EM waveforms. Source magnitudes areproportional to the EM property contrasts (current conductivity,electric permittivity, magnetic permeability) associated with the targetmaterial (proppant in the fracture). The EM wavefield radiated fromthese sources (often referred to as “Born scattering sources”) isrecorded by sensors located at the field data acquisition positions. Animportant consideration for both FBA modeling runs is that a goodestimate of the 3D EM earth model supporting electromagnetic wavepropagation is available. Any preferred numerical modeling methodology(i.e., finite-differences, finite-elements, discontinuous Galerkin,Green functions, layered media propagators, etc.) may be utilized tocalculate EM data. The fracture imaging approach described here is notdependent upon a particular numerical technique.

FBA modeling directly simulates the difference between EM data recordedby the two field experiments. Of course, accurate agreement betweencalculated and observed data is not initially expected for two reasons:i) the 3D numerical earth model supporting EM wave propagation differs(hopefully only slightly) from the 3D true earth model, and ii) thestrengths of the Born scattering sources located in the target volumeare not known sufficiently precisely. In this step of the fractureimaging procedure, the strengths of these sources are adjusted untilcalculated and observed EM data agree to within a prescribed tolerance.A common measure of data misfit is a weighted least squares differencebetween the observed EM time series data and the analogous datasimulated by the FBA modeling approach. Recall that in this context, theword “data” refers to the difference in EM time series data recorded bythe pre- and post-fracture experiments. The described procedureconstitutes a geophysical inverse problem, whereby the strengths of theBorn scattering sources are quantitatively inferred by minimizing thedata misfit. In fact, this is a linear inverse problem that is solvedvia relatively simple methods from linear algebra. The size or “scale”of the inverse problem is determined by the number of Born scatteringsources (equal to the number of multi-component EM receivers distributedover the target volume), which may range into the low thousands.

After the strengths of the many Born scattering sources are determined,an image of the proppant-filled fracture is obtained by visualizing(with any preferred visualization software) the 3D distribution of theseBorn scatterers. The visual image (or map, or picture) isamplitude-calibrated to distinguish the spatially-variable scatteringstrengths. The visual image may be amplitude calibrated using a colorplotting scheme. These strengths are proportional to the contrast in EMproperties (current conductivity, electric permittivity, and magneticpermeability) possessed by the injected proppant relative to thesurrounding geologic medium. In effect, the fracture is defined by the3D distribution of amplitude-calibrated Born scattering sources. If theproppant does not have sufficient parameter contrast, or if it notinjected into remote parts of the hydraulic fracture, then it will notbe imaged via this technique. The disclosed procedure may be used toinfer the spatial extent of proppant fill within a fracture.

Other aspects will be appreciated upon reading and understanding theattached figures and description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an exemplary diagram of a well system that is configured toextract natural gas or oil from a geologic formation beneath the surfaceof the Earth in accordance with an embodiment.

FIG. 2 is an exemplary diagram of a portion of the well system of FIG. 1showing the locations of Born scatters in and around the geologicformation prior to a hydraulic fracturing operation in accordance withan embodiment.

FIG. 3 is an exemplary diagram of a portion of the well system of FIG. 1showing the locations of the same Born scatters (as in FIG. 2) in andaround the geologic formation following a hydraulic fracturing operationin accordance with an embodiment.

FIG. 4 is an exemplary diagram of a set of Born scatterers that arearranged in a three-dimensional grid in accordance with an embodiment.

FIG. 5 is a functional block diagram of an exemplary system thatfacilitates computing a location and an image of a proppant in afracture in a geologic formation beneath the surface of the Earthshowing how data may be transferred and computed by the system inaccordance with an embodiment.

FIG. 6 is a flow diagram that illustrates an exemplary process forcomputing a location, length, orientation and/or image of proppantwithin a fracture in a geologic formation beneath the surface of theEarth in accordance with an embodiment.

FIG. 7 is a flow diagram that illustrates further details of theexemplary process of FIG. 6 in accordance with an embodiment.

FIGS. 8A and 8B combine to show a flow diagram that illustrates furtherdetails of the exemplary process of FIG. 7 in accordance with anembodiment.

FIG. 9 is an exemplary computing system in accordance with anembodiment.

DETAILED DESCRIPTION

Various technologies pertaining to modeling a fracture in a geologicformation will now be described with reference to the drawings, wherelike reference numerals represent like elements throughout. In addition,several functional block diagrams of exemplary systems are illustratedand described herein for purposes of explanation; however, it is to beunderstood that functionality that is described as being carried out bycertain system components may be performed by multiple components.Similarly, a component may be configured to perform functionality thatis described as being carried out by multiple components. Additionally,as used herein, the term “exemplary” is intended to mean serving as anillustration or example, and is not intended to indicate a preference.

As used herein, the term “component” is intended to encompasscomputer-readable data storage that is configured withcomputer-executable instructions that cause certain functionality to beperformed when executed by a processor. The computer-executableinstructions may include a routine, a function, or the like. It is alsoto be understood that a component or system may be localized on a singledevice or distributed across several devices.

With reference now to FIG. 1, an extraction system (system) 100according to an embodiment of the disclosure is disclosed. Theextraction system 100 is configured to extract natural gas, oil orgeothermal resources by way of an induced fracture 117. The system 100includes a well bore 102, which extends from the Earth's surface 104 toa subsurface geologic formation (formation) 106 that contains oil,natural gas or geothermal resources. While the well bore 102 is shown asbeing vertical in nature, it is to be understood that the well bore 102and/or the formation may be vertical, horizontal, dipping, diagonal,slanting or any combination of these. As is well known, the well boremay extend generally vertical to reach the subsurface formation and thenturn horizontal to extend horizontally or laterally through theformation. In such a configuration, the induced fracture 117 may extendvertically and/or horizontally outward from the well bore 102. In anexemplary embodiment, the formation 106 may be several thousand feetbelow the surface 104 of the Earth. Formation 106 may, for example,consist of shale rock. A well casing 108 may be positioned in the wellbore 102 and extend from the surface 104 to or through to below theformation 106. The well casing 108 can be installed in the well bore 102through utilization of any suitable method. Typically, the well casing108 is formed of steel. A cement stabilizer 110 may be formed tostabilize the well casing 108 in the well bore 102. The cementstabilizer 110 stabilizes the casing 108 as fracture fluid and/or aproppant is transferred to the formation 106, possibly under highpressure. The cement stabilizer 110 can also stabilize the well casing108 as natural gas, oil or thermal fluids are extracted from thegeologic formation 106 by way of the well bore 102.

Through utilization of a fracturing fluid under high pressure, afracture 117 including first portions 118 and second portions 120 isinduced in the formation 106. In this exemplary embodiment, the fracture117 is shown simplified as first and second portions 118, 120, however,it should be understood that the fracture 117 may contain several ormultiple fractures, extending horizontally, vertically, and at variousangles, and separate or branching from other induced fractures andcombinations thereof. The fracture may extend laterally and verticallysome distance in all directions from well bore 102. A proppant 119 isdirected down the well bore 102 and fills or partially fills the firstportions 118 of the fracture 117, thereby causing the first portions 118to remain open (and thus causing the formation 106 to be more permeablefor fluid flow). The proppant 119 filling the first portions 118 may bereferred to as a “proppant pack” filling the first portions 118 of thefracture 117. The second portions 120 of the fracture 117 are not filledby proppant 119 and are typically filled with water, sand, gas and/orother rock particles from the surrounding formation 106.

An electric current source 112, which typically resides on the Earth'ssurface 104 is coupled to the casing 108 at a current injection (orcurrent application) point 116 (e.g., positioned near the bottom of wellbore 102 in contact with casing 108 proximate to the geologic formation106 and the proppant-filled fracture 118). In another embodiment, theelectric current source 112 may reside on or below the surface. Inanother embodiment, the current injection point 116 may be locatedwithin geologic formation 106, but not in contact with fracture portion118, or it may be located entirely outside geologic formation 106.Electric current is carried from the current source 112 to the injectionpoint 116 via an insulated wire 114 within the well bore 102.Alternately, the insulated wire 114 may be located on the exterior ofthe casing 108 (i.e., between casing 108 and cement 110). In stillanother embodiment, the electric current source 112 may be locatedwithin well bore 102 proximate to the current injection point 116. Theelectric current source 112 may be configured to generate currentwaveforms of various types (i.e., pulses, continuous wave, or repeatingor periodic waveforms). Accordingly, the well casing 108 can beelectrically energized and act as a spatially-extended source ofelectric current.

Some of the electric current generated by the source 112 can travel fromthe well casing 108 through the proppant 119 of the induced fracture 117of the geologic formation 106. Electromagnetic fields generated by thecurrent in the well casing 108 and that propagate to various locationsin a volume of Earth can be altered by the presence of the proppantfollowing the injection of the proppant 119 into a fracture 117.

The proppant 119 can be chosen to have electromagnetically suitableproperties for generating, propagating, and/or scatteringelectromagnetic fields that can be detected at the Earth's surface 104.For example, the proppant 119 may be chosen to have a particularelectric permittivity, magnetic permeability, current conductivity,and/or other electromagnetic or mechanical properties that are differentfrom the corresponding properties of the surrounding rock of formation106. In this way, a first portions 118 of a fracture 117 that is filledwith proppant 119 will have different electromagnetic properties fromthe second portions 120 of the fracture 117 not filled with proppant119, as well as the rock of the surrounding geologic formation 106. Theproppant 119 can, for example, be formed from an electrically conductivematerial to significantly enhance the electric conductivity of the firstportions 118.

In one embodiment, all of the proppant that is injected into the wellbore and the fracture can be formed from the conductive proppantmaterial. However, this is merely illustrative. In various embodiments,the proppant material can include portions having differentelectromagnetic properties in different portions of the well bore and/orthe fracture. For example, in some circumstances it may be desirable tohave conductive proppant in one portion of a fracture (e.g., a portionof the fracture that is furthest from the well bore or a portion of thefracture that is nearest to the well bore) and non-conductive proppantin another portion of the fracture or in the well bore. In anotherexample, in may be desirable to have proppant material with continuouslyor discretely varying electromagnetic properties as a function of theposition of the proppant material in the fracture.

Providing proppant material having differing electromagnetic properties(e.g., non-conductive and conductive proppant) into a fracture mayinclude mixing conductive materials of differing concentrations into theproppant material as it is injected into the well bore in continuouslyor discretely varying time intervals or may include first injectingconductive proppant into the well bore followed by injectingnon-conductive proppant (as examples). In an embodiment, the proppantmay include both conductive and nonconductive proppant materials. Forexample, the first five, ten, or twenty percent of the proppant materialthat is provided into the well bore may be conductive proppant and theremaining ninety-five, ninety, or eighty percent of the proppantmaterial that is provided into the well bore may be non-conductiveproppant so that only the fracture (or only a leading portion of thefracture) may be filled with the conductive portion of the proppantmaterial. It should be appreciated that these examples are merelyillustrative and that in general any electromagnetically suitableproppant material can be provided.

Electric current source 112 situated on the Earth's surface 104generates electric current that flows down the insulated wire 114 to thecurrent injection point 116 proximate to geologic formation 106 and theproppant-filled first fracture portions 118 contained therein. As theinjection point 114 is in direct physical contact with the well casing108 and the proppant-filled first fracture portions 118, electriccurrent can flow from injection point 116 to the conductive well casing108 and the conductive proppant-filled fracture first portions 118.Current flow within well casing 108 is generally vertically upwards anddownwards from injection point 116, whereas it is laterally outwardsinto geologic formation 106 within the proppant-filled first fractureportions 118. Electromagnetic fields 122 generated by the currents inboth the well casing 108 and the proppant 119 propagate to variouslocations in a three-dimensional volume of Earth. In another embodiment,the electric current source may be located on or below the Earth'ssurface.

Electric currents associated with the electromagnetic waves 122 flowgenerally toward the current grounding point 124 situated on the Earth'ssurface 104. In another embodiment, grounding point 124 may be locatedon or slightly beneath the Earth's surface near to or far from the wellbore 102. In another embodiment, grounding point 124 can be locatedbeneath the surface 104 in another borehole that is relatively near toor far from the well and extraction system 100 and/or geologic formationsystem 106. The another borehole may or may not be used in thefracturing process. Grounding point 124 is connected to the electriccurrent source 112 via the insulated wire 126. In this manner, theinsulated wire 114, current injection device 116, well casing 108,proppant-filled fracture first fracture portions 118, electromagneticwaves 122 propagating within the Earth, grounding device 124, andinsulating wire 126 constitute a “closed loop” that carries electriccurrent from and ultimately back to the electric current source 112. Inan embodiment, the insulated wire 114 may be shielded.

One or more sensors, such as sensors 128, are positioned on the surface104 of the Earth. In another embodiment, one or more sensors 128 may bepositioned on, above or below the surface 104. Sensors 128 are used todetect electromagnetic fields such as the electromagnetic waves 122 thatpropagate from the energized well casing 108 and the proppant-filledfracture first portions 118 to the sensors 128. Sensors 128 include atransducer (not shown) for sensing an EM wave. The sensors 128 mayinclude one or more antennas and receiver circuitry for transmitting,processing, digitizing, or otherwise handling electromagnetic fielddata.

Sensors 128 may be located at corresponding locations such as sensorlocations L1 and L2. Sensors 128 may be deployed in a one-, two-, orthree-dimensional distribution at or near surface 104. For example,sensors 128 may be positioned on surface 104, beneath surface 104 and/orsuspended or mounted above surface 104. Additionally, sensors 128 may bedeployed in various other subsurface boreholes located near to, or atsome distance away from, the geologic formation 106. In variousembodiments, the optimal locations of sensors 128 for detectingelectromagnetic fields can be determined through numerical modeling.Sensors 128 may include various types of physical transducersappropriate for detecting electric fields and/or magnetic fields, andconverting these physical signals to voltage that are subsequentlyforwarded to the data recording system 130. In particular, sensorscommonly used for geophysical exploration or characterization purposes(e.g., porous pots, metal electrodes, electric/magnetic pickup coils,antennas) may be used.

Sensors 128 are connected to the data recording system 130. The datarecording system 130 has the capability to receive, amplify, filter,digitize, process, and otherwise handle the voltage signals generated bysensors 128 in response to the incident electromagnetic waves 122.Additionally, data recording system 130 may store these digitized andprocessed signals on an appropriate recording medium contained therein.Alternately, the data recording system 130 may transmit the receivedsignals to computing equipment 132 where additional processingoperations may be conducted and the data are stored therein. Thecomputing equipment 132 may be located proximate to the data recordingsystem 130, or it may be situated in a remote location. Transmission ofdata between the recording system 130 and computing equipment 132 may bevia an electrical wire, or via radio-transmission techniques.

In certain embodiments, a sensor 128, a data recording system 130, andcomputing equipment 132 may be incorporated into a single physicalpackage or unit capable of being deployed either on the Earth's surface104, or within a subsurface borehole. In this manner, the separatefunctions of signal transduction, amplification, filtering, digitizing,processing, etc. and storage are contained within one physical device

Computing equipment 132 may be used to store a geophysical/geologicalmodel representing the three-dimensional volume of the Earth supportingthe propagating electromagnetic waves 122 (which includes the particulargeologic formation 106 containing the fracture 117). It may also storedata corresponding to the known location of the current injection point116, as well as the known amplitude and waveform of the electric currentgenerated by the current source 112. It may also store the knownthree-dimensional configuration of the well bore 102 with associatedcasing 108 and cement 110, and the known locations of theelectromagnetic sensors 128. It may also store data corresponding to thelocations of a number of Born scatterers in a subsurface volume of theEarth.

Computing equipment 132 may also store numerical algorithms appropriatefor calculating various electromagnetic fields, including those incidenton surface sensors 128 (as with the electromagnetic waves 122), incidenton subsurface Born scattering locations, or scattered by thesesubsurface Born scatterers. Computing equipment 132 may also possessnumerical algorithms and for receiving electromagnetic field data fromsensors 128 and/or the recording system 130, adjusting the parameters ofthe First Born Approximation model of the subsurface using thesereceived electromagnetic data, and determining the location and geometryof the proppant 119 within the fracture 118 using these adjustedparameters. Computing equipment 132 may be used to generate an image ofthe proppant-filled portions 118 of the fracture 117 using appropriatevisualization software, by plotting in three-dimensional space themagnitudes of the Born scatterers. Computing equipment 132 may performall suitable computing, analysis, numerical simulation, data processing,and visualization functions associated with a Born Scattering Inversion(BSI) procedure for imaging the proppant-filled portions 118 of fracture117.

Sensors 128 may be used to gather electromagnetic field data before,during, and after the hydraulic fracturing and proppant injectionoperations. Equipment such as drilling and extraction equipment 134 forcreating, reinforcing, pumping, extracting or other drilling and/orextraction operations may be present in the vicinity of the well bore102. The locations of computing equipment 132 should be maintainedduring gathering of all electromagnetic field data by sensors 128 sothat electrically conductive equipment does not move or changeoperations, and thus undesirably alter the electromagnetic fields to bemeasured. In this way, changes in measured electromagnetic fields beforeand after the hydraulic fracturing and proppant insertion operations canbe primarily or completely attributed to the presence of theproppant-pack 119 within the fracture, thereby increasing the likelihoodthat the First Born Approximation is applicable to the scatteredelectromagnetic fields.

It can therefore be ascertained that by electrically energizing the wellcasing 108 (via the current insertion device 116) to cause it to act asa source of electric current, an electromagnetic field can be inducedand recorded by sensors 128 at the surface 104 of the Earth. Theserecordings, taken before and after fracturing and proppant insertion,can subsequently be used to adjust the parameters of a First BornApproximation model of the scattered electromagnetic field, and therebyindicate the location and geometry of the proppant-filled portion 118 ofthe fracture 117 within geologic formation 106. As used herein, the term“geometry” can refer to the size, shape, length, height, width,orientation, etc. portions of the proppant-filled portions 118 of thefracture 117. “Orientation” can refer to the orientation of at least aportion of the proppant-filled fracture 118 relative to the surface 104or the well bore 102 in the subsurface. The term “location” can refer tothe position of the fracture portion 118 relative to the surface 104,the well bore 102, and/or the current injection point 116.

With reference now to FIG. 2, a portion of geologic formation is shownprior to a hydraulic fracturing operation. As shown in FIG. 2, one ormore Born scatterers 200 may be defined in a geophysical model of theEarth at various locations (e.g., scatterer locations L1′, L2′, etc.) ina volume of the Earth in and around the geologic formation 106.Scatterer locations such as locations L1′ and L2′ can be chosen based ona geophysical model of the geologic formation and any a priori knowledgeof the intended fracturing operation. Scatterer locations can be chosenat any suitable number of locations in and around geologic formation106. As examples, one scatterer location, two scatterer locations, morethan two scatterer locations, ten scatterer locations, one hundredscatterer locations, one thousand scatterer locations, more than onethousand scatterer locations, tens of thousands of scatter locations,between a hundred and a thousand scatter locations or any other suitablenumber of scatterer locations can be chosen. At each scatterer location,a Born scatterer may be defined for use in the first Born approximationmodel.

In an electromagnetic field modeling operation using a first Bornapproximation model such as a Born Scattering Inversion operation, ageophysical model of the volume of the Earth in the location of the wellbore, geologic formation and fracture is used to calculate primaryelectromagnetic fields (sometimes referred to herein as incidentelectromagnetic fields) at the scatterer locations. In an embodiment,the geophysical model may include the location of the well casing andelectricity source. In a first modeling run, the Born scatterers areconsidered to be receivers of the primary electromagnetic fields.

According to the FBA model, a perturbation in the electromagneticproperties of the Earth at the locations of the Born scatterers willgenerate secondary electromagnetic fields (sometimes referred to hereinas scattered electromagnetic fields). Because electromagneticallysuitable proppant is provided in a fracture, a fracture can be modeledas a perturbation in the electromagnetic properties of some of the Bornscatterers.

As shown in FIG. 3, when a fracture 117 is induced in the geologicformation, the fracture 117 may pass through some of the scattererlocations 200. Scatter locations 200 include scatter locations in theproppant filled fracture 200P and scatter locations outside of theproppant filled fracture 200F. Proppant having electromagneticproperties (e.g., electric permittivity, magnetic permeability, and/orcurrent conductivity) that are different from the correspondingproperties of the geologic formation may be located at the locations ofthose scatterers 200P.

The EM properties at scatterer locations 200P will change significantlyafter the introduction of proppant into the fracture. However, the EMproperties at scatterers 200F (located at the fracture, but away fromthe proppant) and at scatterers 200 (located away from the fracture),will change less than the change of the properties at scatterers 200P,or may not change at all.

As described above in connection with FIG. 1, the first Bornapproximation is an approximation in which the scattered electromagneticfield is proportional to the change in the electromagnetic properties atthe Born scatterers and the incident electromagnetic fields on thoseBorn scattering locations.

Therefore, given the primary (or incident) electromagnetic fields at thelocations of the Born scatterers and an initial estimate of theelectromagnetic properties of the material at those locations(represented, for example, by a set of initial values of the adjustableparameters), secondary (or scattered) electromagnetic fields generatedby the Born scatterers in response to the primary electromagnetic fieldscan be calculated. Secondary electromagnetic fields at the locations ofsensors 128 can be generated by, for example, summing the contributionof all Born scatterers in and around the geologic formation 106.

Two sets of measured electromagnetic field data from the sensors areacquired (e.g., before and after a hydraulic fracturing operation) forcomparison to the calculated scattered fields. The difference in themeasured electromagnetic field data from the two measurements may beattributable solely or primarily to the presence of the proppant-filledfracture. The difference in the measured electromagnetic field data cantherefore be used to adjust the calculated scattered electromagneticfield by adjusting parameters corresponding to the electromagneticproperties of the Born scatterers.

Measured (observed) electromagnetic field data gathered at a particulartime t may include a measured voltage difference ΔV(x_(s),t) and/ortime-derivative of magnetic induction ∂b(x_(s),t)/∂t, observed at a setof sensor locations x_(s) (e.g., a three-dimensional position vectorrepresenting a sensor location such as one of sensor locations L1 or L2of FIG. 1), from which an electric field vector e_(m)(x_(s),t) and/or ameasured magnetic field vector h_(m)(x_(s),t) may be determined orinferred.

Calculated primary (or incident) electromagnetic field data may includea calculated primary electric field vector e_(p)(x_(B),t), a magneticfield vector h_(p)(x_(B),t), or both, computed at a set of Bornscatterer locations x_(B) (e.g., a three-dimensional position vectorrepresenting a scatterer location such as one of scatterer locations L1′or L2′ of FIG. 2).

Calculated secondary (or scattered) electromagnetic field data mayinclude an electric field vector δe_(p),t(x_(s),t), a magnetic fieldvector δh_(p)(x_(s),t), or both, computed at the set of sensor locationsx_(s) (e.g., a three-dimensional position vector representing a sensorlocation such as locations L1 or L2 of FIG. 1). The secondaryelectromagnetic field data are determined by computing effective EM bodysources at each scatterer location x_(B), using the primary electricfields e_(p)(x_(B),t), magnetic fields h_(p)(x_(B),t), or both and a setof adjustable parameters representing the change in the electromagneticproperties of the Born scatterer at that scatterer location X_(B). Theeffective body sources and the set of adjustable parameters may be usedto compute the secondary electromagnetic fields by inserting theeffective body sources and the set of adjustable parameters into anysuitable EM forward modeling algorithm modified for the First BornApproximation. Suitable forward modeling algorithms may include thewell-known “EH” system of partial differential electromagnetic waveequations, solved via explicit time-domain finite-difference techniques,or any other appropriate numerical solution methodology. Other suitableforward modeling algorithms may include Green function or potentialformulations, appropriately modified for the FBA.

The set of adjustable parameters may include a change in currentconductivity δσ(x_(B)), a change in electric permittivity δϵ(x_(B)),and/or a change in magnetic permeability δμ(x_(B)) at each scattererlocation x_(B) or a change in all three or any combination of theseparameters.

The effective body sources may include a current density vectorδj(x_(B),t), a magnetic induction vector δb(x_(B),t), and/or an electricdisplacement vector δd(x_(B),t) at each scatterer location x_(B). In theFBA model, the effective body sources (and therefore the secondaryelectromagnetic fields generated by the effective body sources) areproportional to the incident electromagnetic field. For example, in theFBA model, the current density vector δj(x_(B),t) is the product of theincident electric field and the change in the current conductivity(e.g., δj(x_(B),t)=δσ(x_(B)) e_(p)(x_(B),t)), the magnetic inductionvector δb(x_(m) t) is the product of the incident magnetic field and thechange in the permeability (e.g., δb(x_(B),t)=δμ(x_(B)) h_(p)(x_(B),t)),and the displacement vector δd(x_(B),t) is the product of the incidentelectric field and the change in the permittivity (e.g.,δd(x_(B),t)=δϵ(x_(B)) e_(p)(x_(B),t)).

Because the electromagnetic properties of the Earth at some of the Bornscatterer locations change after insertion of proppant into thefracture, the secondary (or scattered) electromagnetic fields, oncecomputed, can be adjusted to match the change in the electromagneticfields observed at the sensor locations. This may be accomplished bymodifying the adjustable parameters in a First Born Approximation modelof the secondary (or scattered) electromagnetic fields by, for example,a linear estimation of the adjustable parameters that produce the bestfit to the observed change in electromagnetic fields.

During a field data acquisition operation, electromagnetic energysources and sensors in both recording measurements may occupy the samepositions. Recording conditions (e.g., electromagnetic energy sourcemagnitudes, waveforms, and orientations; electromagnetic receiversensitivities and orientations, recording system amplifiers, filters andsampling characteristics, etc.) may also be maintained as closely aspossible between the two data collections.

In some circumstances, the sensors may be moved from a first set ofsensor locations to a second set of sensor locations for the gatheringof electromagnetic field data before the fracturing operation and afterthe proppant has been provided into a fracture. In these circumstances,mathematical methods may be used to translate, rotate, interpolate, orotherwise determine measured and/or modeled electromagnetic fields atthe first and/or second sets of sensor locations or to estimate measuredand/or modeled electromagnetic fields at one of the first or second setsof sensor locations based on determined modeled and/or measuredelectromagnetic fields at the other of the first and/or second sets ofsensor locations. However, this is merely illustrative. In variousembodiments, the sensor locations remain the same for all measurementsof electromagnetic fields and all determinations of modeledelectromagnetic fields in order to reduce the computational burden ofthe modeling operation and to help ensure that the change in theelectromagnetic fields is primarily due to the introduction of theproppant.

In some embodiments, a contribution to the secondary electromagneticfields from each Born scatterer may be individually computed in aseparate modeling run. However, this is merely illustrative. In someembodiments, contributions to the secondary electromagnetic fields fromeach of several groups of Born scatterers may be computed in acorresponding modeling run. In one embodiment, the contribution of allof the Born scatterers may be computed in a single modeling run. Thetotal secondary electromagnetic field at each sensor location is the sumof the contributions of each Born scatterer.

The examples of FIGS. 2 and 3 are cross-sectional diagrammatic views.However, it should be appreciated that Born scatterers 200 can bedistributed in a three-dimensional manner in the volume of Earth in andaround the geologic formation.

FIG. 4 shows an example of a three-dimensional distribution of Bornscatterers. In the example of FIG. 4 Born scatterers 200 are distributedin a regular three-dimensional grid in a given volume at scattererlocations (x_(B), y_(B), z_(B)), sometimes collectively referred to as avector x_(B). However, this is merely illustrative. Born scatterers maybe distributed in any suitable pattern for modeling the location ofproppant and/or imaging the proppant in a fracture in, for example, aBorn Scattering Inversion operation. The volume within which scatterers200 are located may be chosen to include a volume of the Earth thatincludes at least a portion of a geologic formation in which a fracturehas been or will be induced.

As shown in FIG. 4, for a particular Born scatterer, an incident(primary) electric field vector 400 (e.g., e_(p)=[e_(x), e_(y), e_(z)])in the coordinate system of FIG. 4) may be computed in, for example, afirst modeling run. In a second modeling run, a current density 402(e.g., δj_(p)=[δj_(x), δj_(y), δj_(z)]) may be computed. The currentdensity 402 may be used to compute a scattered (secondary) electricfield at the location of one or more of sensors 128 of FIG. 1.

With reference now to FIG. 5, an embodiment of a data flow andcomputation system 5000 is shown for determining the location ofproppant and/or imaging the proppant in a fracture in a geologicformation. As shown in FIG. 5, the system 5000 includes a data store500, a primary field modeling engine 502, a secondary field modelingengine 504, a measured data processing engine 510, a data fitting engine512, and an image generation engine 514.

Data store 500, primary field modeling engine 502, secondary fieldmodeling engine 504, measured data processing engine 510, data fittingengine 512, and image generation engine 514 may be implemented on commoncomputing equipment or one or more separate installations of computingequipment. In one embodiment, primary field modeling engine 502,secondary field modeling engine 504, and some or all of data store 500may be located remotely from a drill site at which a well bore islocated, and measured data processing engine 510, data fitting engine512, and image generation engine 514 may be located at the drill site(e.g., incorporated into computing equipment 132 depicted in FIG. 1).However, this is merely illustrative. In various embodiments, data store500, primary field modeling engine 502, secondary field modeling engine504, measured data processing engine 510, data fitting engine 512, andimage generation engine 514 may be included in computing equipment 132(see FIG. 1), or implemented on any suitable computing equipment, nearto or remote from the drill site.

As shown in FIG. 5, primary field modeling engine 502 and secondaryfield modeling engine 504 may be communicatively coupled to data store500. If desired, other portions of system of FIG. 5 may also becommunicatively coupled to data store 500. For example, primary fieldmodeling engine 502, secondary field modeling engine 504, and/or anyother portion of the system of FIG. 5 may have read and/or write accessto memory and information stored on data store 500.

Data store 500 may be used to store a geophysical model 501, an electriccurrent source model 503, one or more Born scatterer locations 505, oneor more sensor types and locations 507, modeled and/or measuredelectromagnetic field data 509 and 511, or other pertinent information,data, numerical algorithms, and/or computer-readable instructions foruse in the system of FIG. 5.

Geophysical model 501 may include stored data that describes thethree-dimensional size, shape, and location of physical structures suchas geologic formations, the Earth's surface, the well bore, the wellcasing, layers of rock, soil, and/or water between the fracturedgeologic formation 106 of FIG. 1 and the surface, drilling andextraction equipment at the drill site, other physical structures andthe electromagnetic properties (e.g., conductivity, permeability,permittivity, etc.) of these structures.

Electromagnetic (EM) data stored on data store 500 may include modeled(i.e., calculated) EM data 509 and/or measured EM data 511. Measured EMdata 511 may be provided to data store 500 from the EM data recordingsystem 508, which may be physically located near the well site, asillustrated in FIG. 1. In addition to modeled and measured data, thedata store 500 may hold executable code representing a First BornApproximation modeling process having a set of adjustable parameters P(e.g., a conductivity change parameter δσ(x_(B)), a permittivity changeparameter δϵ(x_(B)), and/or a permeability change parameter δμ(x_(B)) ateach scatterer location x_(B)).

When executed by one or more processors using geophysical model 501,electric current source model 503, scatterer locations 505, and sensorlocations 507, an electromagnetic forward modeling engine may providecalculated (or modeled) electromagnetic field values as functions oftime t at selected locations. Electromagnetic field values may becomputed by primary field modeling engine 502 and/or the secondary fieldmodeling engine 504 using any suitable numerical modeling approach(e.g., a finite-differences process, a finite-elements process, adiscontinuous Galerkin process, a Green function process, a layeredmedia propagator process, or any other suitable numerical computationprocess).

Primary field modeling engine 502 is used to calculate primaryelectromagnetic (EM) field data 516 (e.g., primary or incidentelectromagnetic field values) at one or more Born scattering locationsx_(B) within a volume of the Earth that includes at least a portion ofthe geologic formation to be fractured. Modeled primary field data 516may include a predicted primary electric field e_(p)(x_(B),t) and/or apredicted primary magnetic field h_(p)(x_(B),t), as functions of time tat scatterer locations x_(B). Modeled primary field data 516 may becomputed using geophysical model 501, electric current source model 503,and scatterer locations 505 stored in data store 500.

Secondary field modeling engine 504 is used to calculate secondaryelectromagnetic (EM) field data 518 (e.g., secondary or scatteredelectromagnetic field values) at one or more sensor locations x_(s)distributed on the Earth's surface (as with sensors 128 in FIG. 1) orwithin the Earth's subsurface. Modeled secondary field data 518 mayinclude a predicted secondary electric field vector δe_(p)(x_(s)t;P)and/or a predicted magnetic field vector δh_(p)(x_(s)t;P) as functionsof time t at sensor locations x_(s). These modeled secondaryelectromagnetic fields are also functions of adjustable parameters Prelated to the strengths of the Born scattering sources. Modeledsecondary field data 518 may be computed using geophysical model 501,electric current source model 503, scatterer locations 505, and sensorlocations 507, and stored in data store 500.

Sensor locations 505 utilized by the primary and secondary fieldmodeling engines represent locations L1 and L2 of sensors 128illustrated in FIG. 1, and which also correspond to the locations of EMsensors 506 in FIG. 5. These sensors are used to measure electromagnetic(EM) field data before and after hydraulic fracturing and proppantinsertion operations.

EM sensors 506 forward measured data to the EM data recording system 508where these data are stored on appropriate recording media. For example,EM sensors 506 are used to gather electromagnetic field data prior to ahydraulic fracturing and proppant insertion, and to gatherelectromagnetic field data after hydraulic fracturing and proppantinsertion. These two measured EM datasets are referenced by letters Aand B in FIG. 5, which may stand for EM data measured after and beforefracturing and proppant insertion, respectively. Measured data mayinclude an electric field vector e_(m)(x_(s),t; A and B) and/or amagnetic field vector h_(m)(x_(s),t; A and B) observed at sensorlocations x_(s), and are collectively referred to as two measured EMdatasets at sensor locations 520. The two sets of measured EM data maybe different due primarily or completely to the addition ofproppant-filled fractures in geologic formation 106 (of FIG. 1).

The two measured EM datasets at sensor locations 520 (referenced byletters A and B) are provided to measured data processing engine 510.Measured data processing engine 510 may receive these datasets via twoalternate routes or pathways. In the first instance, EM data recordingsystem 508 forwards these data directly to measured data processingengine 510 in real time or near-real time as the EM data are measured.In the second instance, EM data recording engine 508 forwards these datato data store 500, where they are held in computational memory asmeasured EM data 511 for later access by the measured data processingengine 510.

Measured data processing engine 510 may be used to perform varioussignal processing operations on the two measured EM data sets A and B inorder to enhance signal quality, suppress noise, etc. Measured dataprocessing engine 506 is then used to generate measured change EM dataat sensor locations 522, based on the two measured EM datasets at sensorlocations 520 (referenced by A and B). The measured change EM data 522may, for example, include a difference (i.e., a subtraction) of datasetsA and B. The measured change data may include an electric vectorΔe_(m)(x_(s),t) and/or a magnetic vector Δh_(m)(x_(s),t). The measuredchange EM data at sensor locations 522 represents the scattered portionof the measured electromagnetic field data 520 generated by the presenceof proppant in the fracture.

Modeled secondary EM data at sensor locations 518 (e.g., calculatedsecondary electromagnetic field data at the sensor locations 506 basedon an FBA modeling approach) and measured change EM data at sensorlocations 522 (e.g., measured scattered electromagnetic field data atthe sensor locations 506) are provided to the data fitting engine 512.Because modeled secondary EM data 518 is generated using an initial setof adjustable parameters P that have not been informed by any measuredinformation about a proppant-filled fracture, the modeled secondary EMdata 518 and the measured change EM data 522 may be different.

Data fitting engine 512 is used to adjust the adjustable parameters(e.g., a set of parameters P corresponding to electromagnetic propertiesof material at the Born scatterer locations x_(B)) until the modeledsecondary EM data at sensor locations 518 has been modified to match themeasured change EM data at sensor locations 522 to within apredetermined or actively determined range. In one embodiment, aweighted least squares difference may be used to quantify the degree ofmisfit between modeled data 518 and measured data 522. The particularset of parameters PFIT that minimizes the weighted least squaresdifference may be solved for using well known procedures from linearalgebra. However, this is merely illustrative. In various otherembodiments, any suitable data fitting procedure may be used to adjustthe set of adjustable parameters P until an adequate fit between themodeled secondary EM data 518 and the measured change EM data 522 isobtained. The particular set of parameters that provides the best fit(i.e., the minimum misfit) constitutes the optimum parameters PFIT.

The optimum set of adjustable parameters 524 (i.e., fitted parameter setPFIT) that results in the modeled secondary EM data 518 that bestmatches the measured change EM data 522 may be provided to imagegeneration engine 514. Image generation engine 514 may use the optimumparameters 524 to determine the location of proppant within a fractureand/or to generate a two or three-dimensional image 526 of the proppantpack. Because the adjustable parameters (P) each are proportional to theamount of change in an electromagnetic property of the material at aBorn scatterer location, the values of the optimum parameters (PFIT)themselves can be used to construct a proppant image 526. The scattererlocations x_(B) serve as effective image pixel (in two-dimensions) orvoxel (in three-dimensions) coordinates. However, this is merelyillustrative. In various embodiments, image generation engine 514 mayperform image processing operations such as scaling, enhancement,smoothing, filtering or other image processing operations to form theproppant image 526.

Image 526 may be generated in near-real time immediately after ahydraulic fracturing operation to provide a drilling manager or teamwith a three-dimensional image of the proppant inserted into thefracture. This image can then be immediately used to guide furtherfracturing and proppant-insertion operations.

FIG. 6 illustrates operations that may be performed by computingequipment, such as described above in connection with FIGS. 1 and 5, tomodel a hydraulic fracture in a geologic formation are shown in FIG. 6.

At block 600 electromagnetic fields in the vicinity of a hydraulicfracture before and after the fracture is generated may be measured. Thehydraulic fracture may include a proppant material havingelectromagnetic properties configured to enhance the detectability ofthe surface electromagnetic fields. Surface electromagnetic fields mayalso be measured during a hydraulic fracturing operation to create thehydraulic fracture. The hydraulic fracture may be generated in ageologic formation in a volume of the Earth.

At block 602, the parameters of an FBA model of the surfaceelectromagnetic fields may be adjusted using the measured surfaceelectromagnetic fields. The parameters may be adjusted by computing adifference between the measured surface electromagnetic fields beforeand after the fracture is generated, comparing modeled electromagneticfields computed using the first Born approximation model to the measuredsurface electromagnetic fields, and adjusting the parameters based onthe comparison. The parameters may each correspond to a change in anelectromagnetic property of a material such as rock or proppant at thelocation of a Born scatterer.

At block 604, the location of proppant such as a proppant pack in thefracture may be determined using the adjusted parameters. The size,shape, and orientation of the proppant pack within a fracture formationmay also be determined using the adjusted parameters.

At block 606, an image such as a three-dimensional image of the proppantpack may be generated using the adjusted parameters.

Further details of operations that may be performed for modeling ahydraulic fracture in a geologic formation are shown in FIG. 7.

At step 700, first measured electromagnetic field values are gatheredusing a plurality of sensors. The plurality of sensors may be located ator near the surface of the Earth as described above in connection with,for example, FIG. 1. Additionally, sensors may be deployed in boreholesin the subsurface of the Earth.

At step 702, a hydraulic fracturing operation is performed.

At step 704, an electromagnetically suitable proppant pack is providedinto the fracture generated by the hydraulic fracturing operation. Theproppant pack may partially or completely fill the fracture. Theproppant material may have an electric permittivity, a magneticpermeability, a current conductivity and/or another electromagneticproperty that is different from the corresponding property of thesurrounding geologic formation. The proppant may be provided such thatthe proppant pack is coupled to a well casing used in the hydraulicfracturing operation and receives electric current from an electricitysource affixed to the well casing.

At step 706, second measured electromagnetic field values are gatheredusing the same plurality of sensors.

At step 708, the change in electromagnetic field values observed at thesensor locations is modeled using a First Born Approximation (FBA)process having adjustable parameters. The adjustable parameterscorrespond to changes in the electromagnetic properties of a set of Bornscatterers in the FBA model.

At step 710, the parameters of the First Born Approximation model areadjusted based on the modeled change field values, and the first andsecond measured field values. For example, the modeled (i.e., predicted)change field values may be adjusted to minimize the difference betweenthe modeled change field values and the measured change field values(i.e., the difference between the first and second measured fieldvalues) by adjusting the FBA model parameters.

At step 712, the location of the proppant pack may be determined basedon the adjusted parameters of the FBA model.

At step 714, an image such as a two- or three-dimensional image of theproppant pack may be generated using the adjusted parameters.

Still further details of operations that may be performed for modelingelectromagnetic fields scattered by a proppant-filled hydraulic fracturein a geologic formation, and forming a three-dimensional image of thatfracture, are shown in FIG. 8.

At step 800, modeled primary (or incident) electric and/or magneticfield values at a set of predetermined locations in a volume of theEarth are determined using a geophysical model. The geophysical modelmay include a geologic formation that is located at least partiallywithin the volume, other layers of the Earth, the Earth's surface, awell bore, a well casing, and/or other geophysical and geologicalfeatures within the volume. The set of predetermined locations may belocations within or near the geologic formation at which correspondingBorn scattering sources have been defined in a First Born Approximation(FBA) modeling approach.

At step 802, modeled secondary (or scattered) electric and/or magneticfield values at the locations of a set of sensors are determined usingthe geophysical model, the modeled primary electromagnetic field valuesat the set of predetermined locations, and a First Born Approximationmodeling process with adjustable parameters.

At step 804, measured first electric and/or magnetic field values aregathered using a plurality of sensors.

At step 806, after the first measured electric and/or magnetic fieldvalues have been gathered, a hydraulic fracturing operation isperformed. The hydraulic fracturing operation may be performed in aportion of the geologic formation within the volume.

At step 808, an electromagnetically suitable proppant pack is providedinto the fracture generated by the hydraulic fracturing operation.

At step 810, following the providing of the proppant into the fracture,second measured electric and/or magnetic field values are gathered usingthe plurality of sensors.

At step 812, difference values may be generated from the first andsecond measured electric and/or magnetic field values. The differencevalues may be determined by, for example, subtracting the secondmeasured values from the first measured values at each sensor. However,this is merely illustrative. In various embodiments, other differencing,correlation, or comparison techniques may be used to determine a changein the second measured electromagnetic fields with respect to the firstmeasured electromagnetic fields.

At step 814, the measured difference values are compared to the modeledsecondary electric and/or magnetic field values from step 802. Comparingthe difference values to the modeled secondary electric and/or magneticfield values may include determining an additional difference betweenthe measured difference values and the modeled secondary electric and/ormagnetic field values for each sensor, computing a sum of the squares ofthese additional differences, computing a weighted sum of the squares ofthese additional differences, computing a sum of absolute values ofthese additional differences, or otherwise quantitatively comparingthese additional differences.

At step 816, adjustable parameters of a First Born Approximation (FBA)model may be adjusted based on the previous comparison 814 of themeasured difference values and the modeled secondary electric and/ormagnetic field values. Adjusting the parameters may include modifyingsome or all of the parameters from initial values in order to modify themodeled secondary (or scattered) electric and/or magnetic fields tomatch the measured difference values to within a given tolerance. Forexample, the parameters may be adjusted to minimize the weighted sum ofthe squares of the differences between the measured difference valuesand the modeled secondary electric and/or magnetic field values for eachsensor. Adjusting the parameters may change the contribution of each ofthe Born scattering sources to the modeled secondary electric and/ormagnetic fields by adjusting the corresponding contrast in one or moreelectromagnetic material properties at the location of each Bornscattering source.

At step 818, the location, size, shape, orientation or other propertiesof the proppant pack in the fracture may be determined based on theadjusted parameters.

At step 820, an image such as a two or three-dimensional image of theproppant pack may be generated using the adjusted parameters. The imagemay be amplitude-calibrated (e.g., using a color plotting scheme) todistinguish the spatially-variable Born scattering strengths representedby the adjusted parameters.

At step 822, the fracture formation and/or the proppant placementresulting from the hydraulic fracturing operation may be evaluated usingthe determined location, size, shape, orientation, and/or the image ofthe proppant pack.

Now referring to FIG. 9, a high-level illustration of an exemplarycomputing device 900 that can be used in accordance with the systems andmethodologies disclosed herein is illustrated. For instance, thecomputing device 900 may be used in a system that supports computingestimates of electromagnetic fields induced by an energized well casingand proppant-filled fracture in a well system using a first Bornapproximation model. In another example, at least a portion of thecomputing device 900 may be used in a system that supports estimating asize, location, length, orientation, and/or image of proppant within aninduced fracture in a geologic formation beneath the surface of theEarth. The computing device 900 includes at least one processor 902 thatexecutes instructions that are stored in a memory 904. The memory 904may be or include RAM, ROM, EEPROM, Flash memory, or other suitablememory. The instructions may be, for instance, instructions forimplementing functionality described as being carried out by one or morecomponents discussed above or instructions for implementing one or moreof the methods described above. The processor 902 may access the memory904 by way of a system bus 906. In addition to storing executableinstructions, the memory 904 may also store computer-implemented modelsof well casing(s) and/or fracture(s), values indicative of an amount ofelectric current applied to a well casing, values indicative of alocation on a well casing where electric current is applied, sensorlocations, scatterer locations, a first Born approximation model havingadjustable parameters, etc.

The computing device 900 additionally includes a data store 908 that isaccessible by the processor 902 by way of the system bus 906. The datastore 908 may be or include any suitable computer-readable storage,including a hard disk, memory, etc. The data store 908 may includeexecutable instructions, computer-implemented models, etc. The computingdevice 900 also includes an input interface 910 that allows externaldevices to communicate with the computing device 900. For instance, theinput interface 910 may be used to receive instructions from an externalcomputer device, a user, etc. The computing device 900 also includes anoutput interface 912 that interfaces the computing device 900 with oneor more external devices. For example, the computing device 900 maydisplay text, images, etc. on a display 930 by way of the outputinterface 912.

Additionally, while illustrated as a single system, it is to beunderstood that the computing device 900 may be a distributed system.Thus, for instance, several devices may be in communication by way of anetwork connection and may collectively perform tasks described as beingperformed by the computing device 900.

In accordance with an embodiment, a system is provided that includes adatabase that stores: a geophysical model of a volume of Earth includinga geologic formation and a well bore, a set of locations within thevolume, an electromagnetic model, and a set of sensor locations; and aprocessor configured to: predict electromagnetic field values at the setof sensor locations using the electromagnetic model and the geophysicalmodel, receive measured electromagnetic field data gathered at the setof sensor locations, and adjust, based on the measured electromagneticfield data, the predicted electromagnetic field values.

In accordance with another embodiment, the processor is furtherconfigured to determine a location of a proppant in a fracture in thegeologic formation using the adjusted predicted electromagnetic fieldvalues.

In accordance with another embodiment, the processor is furtherconfigured to generate an image of the proppant using the adjustedpredicted electromagnetic field values.

In accordance with another embodiment, the electromagnetic modelincludes a first Born approximation model having a plurality of Bornscatterers at the set of locations within the volume.

In accordance with another embodiment, the processor is configured topredict the electromagnetic field values by computing primary electricfield values at the set of locations within the volume and computingsecondary electric field values at the set of sensor locations using theprimary electric field values at the set of locations within the volume.

In accordance with another embodiment, the first Born approximationmodel includes a plurality of adjustable parameters each correspondingto an electromagnetic property of a material at one of the set oflocations within the volume.

In accordance with another embodiment, the processor is configured toadjust the predicted electromagnetic field values by adjusting theplurality of adjustable parameters.

In accordance with another embodiment, the system further includes aplurality of sensors at the set of sensor locations and processor isconfigured to receive the measured electromagnetic field data from theplurality of sensors.

In accordance with another embodiment, the measured electromagneticfield data includes electromagnetic field data gathered before and aftera hydraulic fracturing operation in the geologic formation using thewell bore.

In accordance with another embodiment, the processor is configured toadjust the plurality of adjustable parameters such that the secondaryelectric field values at the set of sensor locations match a differencebetween the electromagnetic field data gathered before and after thehydraulic fracturing operation to within a predetermined range.

In accordance with an embodiment, a method is provided that includesdetermining a plurality of scattered electromagnetic field values usinga model having adjustable parameters; performing a hydraulic fracturingoperation to create a fracture in a geologic formation; providing anelectromagnetically suitable proppant into the fracture; gathering,prior to the hydraulic fracturing operation, a plurality of measuredelectromagnetic field values at a first plurality of sensor locations;gathering, with the electromagnetically suitable proppant in thefracture, an additional plurality of measured electromagnetic fieldvalues at a second plurality of sensor locations; determining adifference between the plurality of measured electromagnetic fieldvalues and the additional plurality of measured electromagnetic fieldvalues; and modifying at least some of the adjustable parameters basedon a comparison of the difference with the plurality of scatteredelectromagnetic field values.

In accordance with another embodiment, the model includes a first Bornapproximation model, the first plurality of sensor locations is the sameas the second plurality of sensor locations, and determining theplurality of scattered electromagnetic field values using the modelhaving the adjustable parameters includes determining a plurality ofprimary electromagnetic field values at a set of predetermined locationswithin a volume of Earth that at least partially includes a geologicformation and determining the plurality of scattered electromagneticfield values at the first plurality of sensor locations using the firstBorn approximation model and using the determined plurality of primaryelectromagnetic fields.

In accordance with another embodiment, the method further includesapplying an electric current to a well bore that extends from a surfaceof the Earth to the geologic formation.

In accordance with another embodiment, modifying the at least some ofthe adjustable parameters based on the comparison of the plurality ofscattered electromagnetic field values with the difference includesdetermining a set of adjusted parameters that minimize an additionaldifference between the difference and the determined plurality ofscattered electromagnetic field values.

In accordance with another embodiment, the method further includesdetermining a location of the proppant within the fracture using the setof adjusted parameters and the set of predetermined locations within thevolume.

In accordance with another embodiment, the first Born approximationmodel includes a plurality of Born scatterers, each located at one ofthe set of predetermined locations within the volume.

In accordance with another embodiment, the method further includesgenerating an image of the proppant within the fracture using the set ofadjusted parameters and the set of predetermined locations within thevolume.

In accordance with another embodiment, the image includes athree-dimensional image of the fracture.

In accordance with an embodiment, a system is provided that includes aconductive well casing in a well bore that extends from a surface ofEarth to a geologic formation within the Earth; an electricity sourceconductively coupled to the conductive well casing; a proppant in afracture in the geologic formation and conductively coupled to the wellcasing; a plurality of sensors at a corresponding plurality of sensorlocations, where the sensors are configured to gather electromagneticfield data generated when a current is applied to the conductive wellcasing using the electricity source; and computing equipment including:memory that stores a plurality of predicted electromagnetic field valuesat the plurality of sensor locations based on a first Born approximationmodel that includes a plurality of adjustable scaling factors; and aprocessor configured to receive the electromagnetic field data from theplurality of sensors, adjust the adjustable scaling factors based on theelectromagnetic field data, and generate an image of the proppant usingthe adjusted adjustable scaling factors.

In accordance with another embodiment the adjustable scaling factorsinclude scattering amplitudes of a corresponding plurality of Bornscatterers.

It is noted that several examples have been provided for purposes ofexplanation. These examples are not to be construed as limiting thehereto-appended claims. Additionally, it may be recognized that theexamples provided herein may be permutated while still falling under thescope of the claims.

What is claimed is:
 1. A system, comprising: a well bore extending fromEarth's surface to a subterranean geologic formation; a casing placedwithin the well bore; an induced fracture extending at leastsubstantially perpendicularly from the casing into the geologicformation; proppant and a conductive material placed within the inducedfracture; an electric current source in electrical communication with atleast a portion of the casing, at least a portion of the conductivematerial and a plurality of sensors at a set of sensor locations; adatabase that stores: a geophysical model of a volume of Earth includingthe geologic formation and the well bore, a set of locations within thevolume, an electromagnetic model, and the set of sensor locations; and aprocessor configured to: predict electromagnetic field values at the setof sensor locations using the electromagnetic model and the geophysicalmodel by: computing primary electric field values at the set oflocations within the volume; and computing secondary electric fieldvalues at the set of sensor locations using the primary electric fieldvalues at the set of locations within the volume, receive measuredelectromagnetic field data gathered at the set of sensor locations,adjust, based on the measured electromagnetic field data, the predictedelectromagnetic field values, determine a location of the proppant in afracture in the geologic formation using the adjusted predictedelectromagnetic field values, and generate an image of the proppantusing the adjusted predicted electromagnetic field values.
 2. The systemdefined in claim 1, wherein the plurality of sensors comprises a firstplurality of sensors configured to detect electric fields and a secondplurality of sensors configured to detect magnetic fields.
 3. The systemdefined in claim 1, wherein at least one of the plurality of sensors islocated within the well bore.
 4. The system defined in claim 1, whereinthe electromagnetic model comprises a first Born approximation modelhaving a plurality of Born scatterers at the set of locations within thevolume.
 5. The system defined in claim 1, wherein computing thesecondary electric field values comprises inserting effectiveelectromagnetic body sources comprising current density vector, magneticinduction vector, or electric displacement vector or combinationsthereof into a system of partial differential electromagnetic waveequations solved via an explicit time-domain finite-differencetechnique.
 6. The system defined in claim 4, wherein the first Bornapproximation model comprises a plurality of adjustable parameters eachcorresponding to an electromagnetic property of a material at one of theset of locations within the volume.
 7. The system defined in claim 6,wherein the processor is configured to adjust the predictedelectromagnetic field values by adjusting the plurality of adjustableparameters.
 8. The system defined in claim 7, further comprising aplurality of sensors at the set of sensor locations, wherein processoris configured to receive the measured electromagnetic field data fromthe plurality of sensors.
 9. The system defined in claim 8, wherein themeasured electromagnetic field data comprises electromagnetic field datagathered before and after a hydraulic fracturing operation in thegeologic formation using the well bore.
 10. The system defined in claim9, wherein the processor is configured to adjust the plurality ofadjustable parameters such that the secondary electric field values atthe set of sensor locations match a difference between theelectromagnetic field data gathered before and after the hydraulicfracturing operation to within a predetermined range.
 11. The system ofclaim 1, further comprising a second plurality of sensors at a secondplurality of sensor locations and configured to gather a plurality ofmeasured electromagnetic field values.
 12. The system of claim 1,wherein the image comprises a three-dimensional image of the fracture.13. The system of claim 1, wherein the proppant comprises non-conductiveproppant and the conductive material comprises conductive proppant. 14.A system, comprising: a well bore extending from Earth's surface to asubterranean geologic formation; an induced fracture extending at leastsubstantially perpendicularly from the well bore into the geologicformation; an electrically conductive proppant pack located within theinduced fracture; an electric current source in electrical communicationwith at least a portion of the induced fracture and a plurality ofsensors at a set of sensor locations; a database that stores: ageophysical model of a volume of Earth including the geologic formationand the well bore, a set of locations within the volume, anelectromagnetic model, and the set of sensor locations; and a processorconfigured to: predict electromagnetic field values at the set of sensorlocations using the electromagnetic model and the geophysical model by:computing primary electric field values at the set of locations withinthe volume; and computing secondary electric field values at the set ofsensor locations using the primary electric field values at the set oflocations within the volume, receive measured electromagnetic field datagathered at the set of sensor locations, adjust, based on the measuredelectromagnetic field data, the predicted electromagnetic field values,determine a location of the electrically conductive proppant pack usingthe adjusted predicted electromagnetic field values, and generate animage of the electrically conductive proppant pack using the adjustedpredicted electromagnetic field values.
 15. The system of claim 14,wherein the plurality of sensors comprises a first plurality of sensorsconfigured to detect electric fields and a second plurality of sensorsconfigured to detect magnetic fields.
 16. The system of claim 14,wherein at least one of the plurality of sensors is located within thewell bore.
 17. The system of claim 14, wherein the electromagnetic modelcomprises a first Born approximation model having a plurality of Bornscatterers at the set of locations within the volume.
 18. The system ofclaim 17, wherein the first Born approximation model comprises aplurality of adjustable parameters each corresponding to anelectromagnetic property of a material at one of the set of locationswithin the volume and wherein the processor is configured to: adjust thepredicted electromagnetic field values by adjusting the plurality ofadjustable parameters, and receive the measured electromagnetic fielddata from the plurality of sensors.
 19. The system of claim 14, whereinthe image comprises a three-dimensional image of the fracture.
 20. Thesystem of claim 14, wherein the electrically conductive proppant packcomprises non-conductive proppant and conductive proppant.